the new intersection theorem and descent of flatness for integral extensions

Journal of Algebra 322 (2009) 3142–3150

www.elsevier.com/locate/jalgebra

The new intersection theorem and descent of flatnessfor integral extensions

Phillip Griffith

Department of Mathematics, University of Illinois, 1409 West Green Street, Urbana, IL 61801, USA

Received 13 December 2007

Available online 1 July 2008

Communicated by Melvin Hochster

In honor of Professor Paul Roberts’ contributions to mathematics

Abstract

The new intersection theorem is used to derive a criteria for flat descent in the setting of integral ringextensions. Applications, such as “purity of branch locus” for extensions of normal domains, are noted.© 2008 Elsevier Inc. All rights reserved.

Keywords: Commutative rings; Integral extensions; Flatness; Galois theory of rings; Purity of branch locus; Lifting

Introduction

The new intersection theorem represents one of the great achievements in the field one mightrefer to as “homological commutative algebra.” It was originally formulated by Peskine andSzpiro [23] as a generalization of their intersection theorem [22, pp. 84–86] when the ambientring is singular. The result was proven by them [23] in the case of positive prime characteristicand in some cases of equicharacteristic zero. Hochster [18] showed that the conjecture had an af-firmative answer when big Cohen–Macaulay modules were known to exist. In 1987 Roberts [25]constructed a remarkable proof of the new intersection theorem in the remaining case of mixedcharacteristic. Roberts’ methods were quite different from those of Peskine and Szpiro [23] andHochster [18]. Namely, his development of special properties of local Chern characters (“theycommute with intersections with divisors”; see [26, pp. 280–284]) led to a beautiful proof in themixed characteristic case. It should be mentioned that Gillet and Soulé [13] independently con-

E-mail address: [email protected].

0021-8693/$ – see front matter © 2008 Elsevier Inc. All rights reserved.doi:10.1016/j.jalgebra.2008.05.005

P. Griffith / Journal of Algebra 322 (2009) 3142–3150 3143

structed a proof of this theorem as a result of their work concerning Adams operations. Simplystated, the new intersection theorem says: If A is a local ring and if F• is a nontrivial boundedfree A-complex whose homology has finite length, then length F• � dimA.

In this article we study a rather different application of the new intersection theorem in thecontext of integral extensions and descent of flatness. For purpose of discussion let R ↪→ A repre-sent an integral extension of Noetherian rings for which R and A satisfy the Serre condition (S2).Let M be a finitely generated R-module having finite projective dimension and also satisfy-ing (S2). Our main result in Section 1 (Theorem 1.1) states that, if HomR(M,A) is A-projective,then M must be R-projective. We remark here that the key idea for using the new intersectiontheorem to prove Theorem 1.1 appears (somewhat buried) in a proof given by Kantorovitz [19,Theorem 2.1] in which she generalized Auslander’s module theoretic proof of “purity of branchlocus” [2] to the setting of module finite extensions of normal domains. The remainder of Sec-tion 1 is devoted to showing how Theorem 1.1 can play a role in achieving more direct proofs,and in some cases more general results, even when the topic is familiar and classical. The settingwe have chosen in which to demonstrate this claim is that of separable/Galois ring extensions(see DeMeyer and Ingraham [11] for a detailed discussion on this topic). The setup for resultson purity of branch locus also fits into our discussion. In either of these contexts one is oftenconfronted with the task (sometimes painful) of establishing that A is R-projective (notation asabove). Once this property has been secured then, for the most part, the remaining obstacles areroutinely removed (e.g., see Auslander and Buchsbaum [3, Theorem 3.87] and their discussionon ramification theory). A classical example is Auslander’s module theoretic proof [2] of Na-gata’s theorem [21] on purity for R (as above) regular local. The proof of the crucial moduletheoretic result [2, Theorem 1.4] requires a long and complex argument. By making use of herversion of Theorem 1.1, Kantorovitz [19] achieves a straightforward and more general proof.

In Section 2 we consider the ring map A → EndR A in case it is flat; the codomain structureis the relevant one in use for EndR A. In addition, we assume the base ring R is a complete localnormal domain and the module finite extension is generically Galois. We impose an additionalassumption that the normal basis property holds for R ↪→ A in codimension � 3 (a much weakerassumption than unramifiedness). We describe in Corollary 2.2 and Theorem 2.3 how these prop-erties lead in a natural way to results on “lifting” A as a left R[G]-module that in turn leads to A

being R-free, even though one does not know A has finite projective dimension (over R) at theoutset. Thus when R is required to be a complete intersection we establish fairly simple criteriaso that A is necessarily Cohen–Macaulay.

0. Remarks on notation, definitions and basic references

For the most part we follow Matsumura’s text [20] in regard to notation and basic defini-tions of terms in commutative algebra. The main references for separable algebras and GaloisTheory of rings follows the language and definitions as given by Auslander and Goldman [5,6]and DeMeyer and Ingraham [11]. Terminology related to purity of branch locus is taken fromAuslander [2], Auslander and Buchsbaum [3] and Kantorovitz [19]. We recall, for a modulefinite extension R ↪→ A, that a prime ideal P ∈ SpecA is said to be unramified over SpecR

means (i) pAP = P Ap and (ii) the induced injection on residue fields k(R/p) → k(A/P ),where p = P ∩ R, is separable. The original theorem on purity of branch locus (see Nagata [21]for a straightforward account) states: for R regular one can decide whether the ring extensionR ↪→ A is unramified at all prime ideals by restricting one’s attention to the prime ideals inSpecA of codimension one.

3144 P. Griffith / Journal of Algebra 322 (2009) 3142–3150

We make reference in Section 2 to normal bases for extensions R ↪→ A (R is local here) ofnormal domains (i.e., integrally closed domains) that are generically Galois with group G. Thissimply means there is an element θ ∈ A such that A = ⊕

σ∈G Rσθ as an R-module. Of coursethis statement is well known to be equivalent to the claim that A is isomorphic to the groupalgebra R[G] as a left R[G]-module. In Chase, Harrison and Rosenberg [9, pp. 27, 28] it isproven: if R ↪→ A is separable and Galois, then R ↪→ A has the normal basis property.

When discussing the notion of lifting modules in the context of regular deformations wefollow the description and spirit of the account given by Auslander, Ding and Solberg [4].

We make one final remark about the module structure given to the homomorphism moduleHomR(M,A). When M is merely an R-module, the A-module structure must be provided by thecodomain “A,” that is, (a f )(x) =: af (x). However, when M is also an A-module there are nowtwo choices for A-module structure. In this article we shall always use the “codomain induced”structure.

1. The main theorem on flat descent

In this section we establish our main result on “descent” that shows certain integral extensionsof rings are necessarily flat extensions.

Theorem 1.1. Let R be a Noetherian ring, let R ↪→ A represent an integral extension for whichA is also Noetherian and let M be a finitely generated R-module. If R, A and M satisfy theconditions

(i) R, A and M are (S2),(ii) pdR M < ∞ (locally over SpecR),

(iii) HomR(M,A) is A-projective,

then M is R-projective.

Corollary 1.2. Suppose part (iii) in the statement of Theorem 1.1 is replaced by (iii)′: the reflexiveclosure of A ⊗R M with respect to A is A-projective. Then the conclusion “M is R-projective”still holds.

Proof of 1.2. The adjoint isomorphism HomA(A ⊗R M,A) ∼= HomR(M,HomA(A,A)) ∼=HomR(M,A) shows that HomR(M,A) is A-projective. Thus we may apply Theorem 1.1 toconclude that M is necessarily R-projective. �Proof of Theorem 1.1. For the argument here we may assume that R is local of dimensiond < ∞. Let F• → M be a minimal R-free resolution of M . The Auslander–Buchsbaum formula,pdR M + depthR M = depthR � d , gives the inequality pdR M � d − 2 and hence that lengthF• = pdR M � d − 2. The induced finite A-complex

(G•) : 0 → HomR(M,A) → HomR(F0,A) → HomR(F1,A) → ·· ·

has length = pdR M + 1 < d = dimA, since pdR M � d − 2. Since the statement of Theorem 1.1is clearly true when dimA = dimR = d � 2 (here M is necessarily R-free since pdR M � 0), wemay assume that d > 2 and that M is locally free on SpecR−mR . It follows that the cohomology,


ExtiR(M,A), of G• is supported at {mR} for each i > 0 (the only indices where nonzero coho-mology can occur). As an A-complex the cohomology is supported at a subset of the maximalideals of A. Locally at each of these maximal ideals we may invoke the Peskine–Szpiro–Robertstheorem (see [26, p. 296]) to see that all of the homology of G• must be zero and that G• is nec-essarily a trivial A-complex. However, should it be the case that pdR M = s > 0, then the matrixthat represents the map HomR(Fs−1,A) → HomR(Fs,A) will have entries in mRA ⊆ radA andtherefore cannot be surjective. Thus we conclude that pdR M = 0 as desired.

Corollary 1.3. Suppose that R ↪→ A is a module finite extension of (S2)-rings, e.g., R ↪→ A isa module finite extension of normal domains, such that the induced ring map A → EndR A isA-flat, when the codomain structure is used for EndR A. If pdR A < ∞ then A is R-projective.

Proof. The statement is an immediate consequence of Theorem 1.1.Classical ways in which the flatness of the endomorphism ring arises is seen in the following

context. Let R be a normal domain and suppose that R ↪→ A is a module finite extension suchthat A is also a normal domain and such that the induced extension of the fraction fields isGalois with group G. Then there is an injective ring homomorphism j :Δ(A;G) → EndR A,where Δ(A;G) represents the “twisted” group algebra (multiplication in Δ(A;G) is defined by:aσ ·a′τ =: aσ(a′)στ ). The injective ring homomorphism j is defined so that j (aσ )(x) = aσ(x),for x ∈ A. In particular, Δ(A;G) is a free A-algebra. In keeping with the classical developmentof field extensions, the notion of Galois ring extension represents a special case of separablering extension. In terms of our current set-up and notation, the Galois requirement means thatthe ring homomorphism j :Δ(A;G) → EndR A is an isomorphism (see [18, pp. 80, 81] and [9,Theorem 1.3]). Thus Galois extensions in the setting of separable R-algebras provide a contextfor which EndR A is naturally A-free. As a consequence of the preceding discussion one obtainsthe next statement which shows that Galois extensions have a codimension one formulation inthis setting.

Corollary 1.4. Let R ↪→ A be a module finite extension of normal domains for which pdR A <

∞. If R ↪→ A is generically Galois, and if R ↪→ A is separable in codimension � 1, then A isR-projective and R ↪→ A is a Galois extension of commutative rings.

Proof. Since the map j :Δ(A;G) → EndR A is an isomorphism in codimension � 1 and sinceA satisfies the condition (S2) we get that j is necessarily an isomorphism. From Corollary 1.3 weconclude that A must be R-projective, and from [11, Proposition 1.2(3)] it follows that R ↪→ A

is a Galois extension of commutative rings.Likewise in Auslander’s module theoretic proof of “purity of branch locus” for regular local

rings R, he reduces to the case R ↪→ A is a module finite extension for which A is normal andR ↪→ A is generically Galois (see [2, p. 118]). The assumption that R ↪→ A is unramified at allprimes of codimension one results in the fact the map j :Δ(A;G) → EndR A is an isomorphismin codimension � 1. It follows as in Corollary 1.4 that j is in fact an isomorphism since A sat-isfies the Serre condition (S2). Once again we have that EndR A is A-free. Since, we have thatpdR A < ∞, hence A is R-free. The remainder of the argument that A/R is étale (flat and unram-ified) then follows easily (e.g., see [3, Theorem 3.8]). Kantorovitz [19] generalized Auslander’sargument, as above, by making use (in an implicit way) of Theorem 1.1. We will state her resultand briefly sketch her argument so that one sees the full power of Theorem 1.1 at work. �


Theorem 1.5. (See Kantorovitz [19].) Let R ↪→ A be a module finite extension of normal do-mains which is unramified in codimension one and suppose pdR A < ∞. Then R ↪→ A is anétale extension.

Proof. In part, the proof [19] employs the Galois closure R ↪→ A ↪→ S where R ↪→ S is gener-ically Galois, S is normal and R ↪→ S is unramified in codimension one (note that one does notknow pdR S < ∞). At this point Kantorovitz makes clever use of some module isomorphisms(see Borek [7, p. 420]) which show that

HomR(A,S)h ∼= HomR(S,S) ∼= Sg,

where the isomorphisms are as S-modules and codomain induced. Here h = [S : A] and g =[A : R]; note if A = S then h = 1. Thus one is now in a position to employ Theorem 1.1 andobtain that A is R-free. Again one finishes the argument along the lines of [3, Theorem 3.8]. �

Our final result of this section can be viewed in the spirit of “splitting rings” for centralseparable algebras (see [6, p. 382] for further discussion and more detail). Let R be a com-mutative Noetherian ring and let Λ be a central separable R-algebra, i.e., Λ is a projectivemodule over its enveloping algebra Λe = Λ ⊗R Λop, and the center of Λ is R. If A is a maximalcommutative subring of Λ such that A is a separable R-algebra, then the ring homomorphismA ⊗R Λop → EndA Λ, given by a ⊗ λ → ε where ε(x) = axλ, is an A-algebra isomorphism. Inparticular Λ is projective as an A-module and thus the class [Λ] represents the zero element inthe Brauer group of A.

Our actual setup here is more relaxed than the above account. Our Λ will be of the formΛ = EndR M and A will be a maximal commutative subring of Λ. We will observe that thecondition A → EndR M = Λ is A-flat, as is the case above, has very strong consequences.

Theorem 1.6. Let R be a normal domain and let M be a reflexive R-module. Let A be a maximalcommutative subring of Λ = EndR M .

(a) If the ring homomorphism A → Λ is flat and pdR A < ∞, then A is R-projective.(b) If in addition to the hypotheses of (a) one has R is regular, then M is R-projective as well.

Proof. It suffices to consider the case where R is local; so A is semi-local. The “maximal”condition on A forces the factor module Λ/A to satisfy (S1) since Λ/A is necessarily R-torsionfree. Hence A must be an (S2)-ring. There is an A-algebra map cited above: θ :A ⊗R Λop →EndA Λ. In codimension one, we have that both Λ ⊗R Λop and EndA Λ are isomorphic to thecentral separable A-algebra Mn(A), where Mn(A) denotes n × n matrices over A. This claimis due to Λ ∼= Mn(R) and Λ ∼= An. It follows that θ is an isomorphism (see [6, Corollary 3.4])in codimension one and that the reflexive closure of the A-module A ⊗R Λop is A-free. SinceA → Λ is necessarily A-split one further obtains that the reflexive closure of A ⊗R A is A-free.By Corollary 1.2 we get that A is R-free.

To justify part (b) we note, in view of the conclusion for part (a), that EndR M is R-free. SincepdR M < ∞ we may use the identical arguments as given in [5, Proposition 4.9 and Theorem 4.4]to argue that M is R-free. �


2. Free endomorphism rings, normal basis property and lifting

Consider a module finite extension B ↪→ A of normal domains. As is well documented(see below) when the condition of finite projective dimension (pdB A) is removed then, gen-erally speaking, no amount of “unramifiedness” for a fixed codimension will ensure that A isB-projective, and hence that B ↪→ A is étale. There is a notable exception to this pathology,namely when B is a complete intersection. Under this assumption, Grothendieck’s theorem [17,Exposé X] concludes that “unramified in codimension three” suffices and Cutkosky’s refine-ment [10] requires only that the extension be “unramified in codimension two” when B is anormal complete intersection. The examples cited in [15, Theorem 4.5] show there can be nocomparable result when B is Cohen–Macaulay or even Gorenstein (the integer “e” in Theo-rem 4.5 [15] can be chosen so that the base ring is Gorenstein). However, the common occurrenceof such extensions in the following context provides some motivation to examine the situation abit further. For example, let R be a regular local ring containing the rational numbers and appro-priate roots of unity. Let R ↪→ D be a module finite extension with D a normal domain. Thenone has the following diagram of module finite ring extensions as described in [14, Section 3]

A

D B

R

in which R ↪→ A, R ↪→ B and B ↪→ A are generically Galois. The diagram is constructed sothat B ↪→ A is unramified in codimension one while the extension R ↪→ B has a cyclic Galoisgroup and B is Gorenstein. Abhyankar [1] first described the above construction in geometriclanguage. From his point of view, one had “confined” the codimension one ramification fromR ↪→ D by isolating it in the cyclic Kummer extension R ↪→ B , where perhaps the “algebra” issomewhat easier to understand. In such a setting the endomorphism ring EndB A is A-free andone has at the very least that B ↪→ A is unramified in codimension one. In case B turns out tobe a complete intersection rather than merely Gorenstein, when can one expect A (and hence D)to be Cohen–Macaulay (i.e., when can one expect A and D to be R-free)? Here we are lookingfor weaker conditions than, say étale. If G = gal(A/B) we note that “B ↪→ A unramified incodimension � i” implies that B ↪→ A has the “normal basis property” in codim � i (see [9,pp. 27–28]). The normal basis property interpreted in terms of the group algebra B[G] meansthat A is isomorphic as a left module to B[G] in codimension � i.

Our first step is to establish a preparatory lemma that examines the module theoretic implica-tions for ExtiΛ(A,A), with j = 0,1,2, when one invokes the normal basis property for B ↪→ A.Here Λ = B[G].

Lemma 2.1. Let B denote a local ring that satisfies the Serre condition (S4) and let B ↪→ A bea module finite extension for which there is a finite group G of B-automorphisms of A such thatAG = B . Let Λ = B[G].

(i) If A satisfies (S1) and if A is a locally free left Λ-module in codimension one over SpecB ,then EndΛ A ∼= Aop.


(ii) If A satisfies the Serre condition (S2) and if A is a locally free left Λ-module in codim � 2then

Ext1Λ(A,A) = 0.

(iii) If A satisfies the Serre condition (S3) and if A is a locally free left Λ-module in codim � 3over SpecB , then ExtjΛ(A,A) = 0 for j = 1,2.

Proof. Part (i) follows from the elementary fact that EndA Λ ∼= Λop (see Rotman [27, p. 529])and that A ∼= Λ as left modules locally in codimension one; thus the natural map Λop → EndΛ A

is an isomorphism in codimension one and therefore an isomorphism since Λ satisfies (S2) as aB-module.

To see part (ii), let x ∈ mR − {0} be a regular element on A and consider the short exactsequence 0 → A

x−→ A → A → 0 and the induced long exact sequence on cohomology

0 → EndΛ Ax−→ EndΛ A → EndΛ A

δ−→ Ext1Λ(A,A)x−→ Ext1Λ(A,A)

→ Ext1Λ(A,A) → Ext2Λ(A,A)

x−→ Ext2Λ(A,A) → ·· · .

The long exact sequence makes sense since x lies in the center of Λ. Making use of part (i)we observe that the first terms of the sequence become

0 → Λx−→ Λ → EndΛ A

δ−→ Ext1Λ(A,A).

Since A is locally free as a left Λ-module in codimension � 2 over SpecB one knows that aminimal prime P in Supp(Image δ) has codimP � 3. Thus localizing at P gives depthΛ � 3,depth EndΛ A � 2 and length(Image δ)p < ∞. It follows from [12, Lemma 1.1] that necessarilyImage δ = 0, that EndΛ A ∼= Λ and that Ext1Λ(A,A) = 0 and

0 → Ext1Λ(A,A) → Ext2Λ(A,A) → Ext2Λ(A,A)

is exact. In case the hypothesis of part (iii) is assumed then a repeat of the above argumentwith A, Λ and x replaced by A, Λ and a regular element y in the center of Λ, respectively, yieldsthat Ext1

Λ(A,A) = 0 and hence that Ext2Λ(A,A) = 0 (check a prime minimal in the support of

Ext2Λ(A,A)). �Corollary 2.2. (Notation as in 2.1.) Suppose that B is a complete local ring and that B ↪→ A

satisfies the hypothesis of Lemma 2.1(iii). If S → B represents a regular deformation of B andΓ = S[G], then A has a unique lifting to a left Γ -module.

Proof. The above statement follows immediately from the criteria for lifting in [4, Proposi-tion 1.6] and the conclusion of Lemma 2.1(iii); the uniqueness is a consequence of [4, Proposi-tion 2.5]. �

Keeping the notation as in Corollary 2.2 we see that the vanishing Ext2Λ(A,A) = 0 means A

will have a lifting to Γ = S[G] in the sense of [4, p. 276]. Here S/x̄S = B where x̄ is a regularS-sequence; so x̄ is a Γ -sequence that lies in the center of Γ . When considering a lifting of the


B-algebra A, normally one would expect to lift its algebra structure as well, that is, one wouldlike an S-algebra T that admits G as a group of S-automorphisms of T such that T G = S andsuch that T/x̄T ∼= A. However, as one can observe in our next results, being able to lift A merelyas a left Λ-module to Γ can have unexpected consequences.

Corollary 2.3. (Notation as in 2.1 and 2.2.) If the ring S in (2.2) is regular local, thenpdB A < ∞.

Proof. Since the deformation is regular, the lift of A to S has finite projective dimension over S

if and only if pdB A < ∞. �The main result of this section follows.

Theorem 2.4. Let B denote a complete local normal domain that is a complete intersection.Suppose B ↪→ A is a generically Galois extension that satisfies

(a) B ↪→ A has the normal basis property in codimension � 3,(b) A has the property (S3),(c) EndB A is A-free (codomain induced structure).

Then A is a free B-module.

Proof. To establish the claim in (2.4) one merely has to observe that B has a regular deformationS in which S is regular local. It follows from Corollary 2.3 that pdB A < ∞. One then appeals toour main theorem (Theorem 1.1) and property (c) of the hypothesis for 2.4 in order to assert thatA must be a free B-module. �2.5 Some final remarks on the hypotheses and proof of Theorem 2.4.

(1) One can see that the hypothesis on normal bases for B ↪→ A cannot be relaxed too muchfrom Example 5.5 [8]. In that example the base ring (denoted by “R” in [8]) is a completeintersection of embedding codimension two and the generically Galois extension (denotedby “S”) fails to be Cohen–Macaulay even though the extension is unramified in codimensionone (so the normal basis property holds in codimension one).

(2) The normal basis property would appear to be significantly weaker than the notation of“étale”. For example, in algebraic number theory, when B = Z, the ring of integers, then onemerely requires that B ↪→ A is tamely ramified by a theorem of Hilbert and Speiser.

3. Note added in proof

To see that the condition “pdR M < ∞” is necessary in the statement of Theorem 1.1, onemay consider a module-finite extension of normal local domains R ↪→ A in which a reflexive R-ideal b represents a nontrivial class in the kernel of the group homomorphism Cl(R) → Cl(A).Here the notation “Cl(R)” denotes the divisor class group of R (see [16] for a definition).Then HomR(b,A) ∼= A (as A-modules), while b �∼= R. To observe a concrete realization, letA = C�X,Y �, where C denotes the field of complex numbers, and let 〈σ 〉 ∼= Z2 act linearly on A

via σ(X) = −X and σ(Y ) = −Y . Then R = Aσ ∼= C�X2,XY,Y 2 � ∼= C�U,V,W �/(W 2 − UV ).


Let b = (x2, xy) where “x” and “y” indicate the residue classes. Then b is a reflexive R-module.However, HomR(b,A) ∼= A since A is a UFD in this case.

We also mention that one should consult Raynaud and Gruson’s article [24] for an extensivediscussion of descent of flatness and projectivity in regard to the tensor functor.

References

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the new intersection theorem and descent of flatness for integral extensions

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